Find the Exact Value cos(19pi)

Problem

cos(19*π)

Solution

1. Identify the periodicity of the cosine function. The cosine function has a period of 2*π meaning cos(θ)=cos(θ+2*n*π) for any integer n

2. Reduce the angle by subtracting multiples of 2*π Since 19*π is an odd multiple of π we can write it as 18*π+π

3. Apply the periodicity rule. Since 18*π is 9×2*π it represents 9 full rotations around the unit circle.

cos(19*π)=cos(18*π+π)

cos(19*π)=cos(π)

4. Evaluate the cosine of the resulting angle using the unit circle. At π radians (or 180, the coordinates on the unit circle are (−1,0) The cosine value is the x-coordinate.

cos(π)=−1

Final Answer

cos(19*π)=−1

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